Solution
= Solution
Use the <product-to-sum formula> and the double-angle <trigonometric identities>:
$$
\cos^3x\sin x=\frac14(1+\cos2x)\sin2x
=\frac14\sin2x+\frac18\sin4x.
$$
By <Fourier orthogonality>, these are already its only nonzero <Fourier series> coefficients:
$$
\boxed{a_n=0\ (n\geq0),\qquad b_2=\frac14,\quad b_4=\frac18,\quad b_n=0\ (n\ne2,4)}.
$$